A significant number of works have set forth, over the past decades, the emphasis laid by seventeenth-century mathematicians and philosophers on motion and kinematic notions in geometry. These works demonstrated the crucial role attributed in this context to genetic definitions, which state the mode of generation of geometrical objects instead of their essential properties. While the growing importance of genetic definitions in sixteenth-century commentaries on Euclid’s Elements has been underlined, the place, uses and status of motion in this geometrical tradition has however never been thoroughly and comprehensively studied. This book therefore undertakes to fill a gap in the history of early modern geometry and philosophy of mathematics by investigating the different treatments of motion and genetic definitions by seven major sixteenth-century commentators on Euclid’s Elements, from Oronce Fine (1494–1555) to Christoph Clavius (1538–1612), including Jacques Peletier (1517–1582), John Dee (1527–1608/1609) and Henry Billingsley (d. 1606), among others. By investigating the ontological and epistemological conceptions underlying the introduction and uses of kinematic notions in their interpretation of Euclidean geometry, this study displays the richness of the conceptual framework, philosophical and mathematical, inherent to the sixteenth-century Euclidean tradition and shows how it contributed to a more generalised acceptance and promotion of kinematic approaches to geometry in the early modern period.
Author(s): Angela Axworthy
Series: Frontiers in the History of Science
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 305
City: Cham
Acknowledgement
Contents
List of Figures
1: Introduction
1.1 Motion in Geometry: A Useful Yet Controversial Notion from Antiquity to the Early Modern Era
1.2 Motion in Geometry and the Sixteenth-Century Euclidean Tradition
1.3 The Significance of Genetic Definitions to the Assessment of the Status of Geometrical Motion
1.4 Sources and Research Questions
1.5 The Uses and Designations of Motion in Ancient Geometrical Texts
1.6 The Ontological Implications of Ancient Kinematic Notions
1.7 Reservations and Objections Against Geometrical Motion from Antiquity to the Middle Ages
1.8 Conditions and Justifications for the Admission of Motion in Geometry
2: Oronce Fine
2.1 The Life and Work of Oronce Fine and the Significance of His Commentary on the Elements
2.2 Formulation and Distribution of Genetic Definitions in the Geometria and in Fine´s Commentary on Euclid
2.3 The Function of Genetic Definitions in Fine´s Exposition of Euclid
2.4 The Ontological Status of Geometrical Motion
2.5 Definitio and Descriptio
2.6 Mathematical Versus Physical Motion
2.7 Motion and the Composition of the Continuum
2.8 The Relation Between the Generation of Magnitudes and the Generation of Numbers in Fine´s Commentary on Book II
2.9 The Metaphysical Interpretation of the Generation of Quantities
3: Jacques Peletier
3.1 The Life and Work of Jacques Peletier and the Significance of His Commentary on the Elements
3.2 Formulation and Distribution of Genetic Definitions in Peletier´s Commentary on Euclid
3.3 The Flow of the Point in Peletier´s Louange de la Sciance
3.4 Peletier´s Metaphysical Discourse in the Commentary on Euclid
3.5 The Motion of the Point and the Propagation of Light
3.6 The Flow of the Point and the Composition of the Continuum
3.7 The ductus of the Line and the Multiplication of Numbers
3.8 Genetic Definitions and the Limits of Human Knowledge
4: François de Foix-Candale
4.1 The Life and Work of François de Foix-Candale and the Significance of His Commentary on the Elements
4.2 Formulation and Distribution of Genetic Definitions in Foix-Candale´s Commentary on Euclid
4.3 Progressus Versus Fluxus
4.4 The Two Definitions of the Sphere: Euclid Versus Theodosius
4.5 The Commentary on Euclid and the Pimandre
4.6 The Ontological Status of the Sphere in the Pimandre
4.7 Foix-Candale and Peletier on the Ontological Status of the Sphere and of the Circle
4.8 Proclus on the Properties and Constitution of the Intelligible Circle
4.9 The Centre of the Sphere as a Divine Principle
4.10 Foix-Candale and Peletier on the Epistemological Status of Geometrical Definitions
4.11 Motion in Foix-Candale´s Commentary on Book II
5: Henry Billingsley
5.1 The Life and Work of Henry Billingsley and the Significance of His Commentary on the Elements
5.2 Motion in Euclidean Geometry According to Billingsley
5.3 The Ontological Status of Geometrical Motion
5.4 The Two Definitions of the Sphere: Essential Versus Causal Definition
5.5 The Description of Geometrical Figures According to Billingsley
5.6 Definition and Description in Foix-Candale and Billingsley
5.7 The Modes of Generation of Numbers and Magnitudes According to Billingsley´s Commentary on Book II
6: John Dee
6.1 The Life and Work of John Dee and the Significance of His Contributions to the Euclidean Tradition
6.2 The Place and Function of Genetic Definitions in Dee´s Teaching of Euclidean Geometry
6.3 The Motion of the Point and Its Ontological Implications
6.4 The Mobility of the Point Versus the Immobility of the Unit
6.5 Human Versus Divine Geometry in the Mathematicall praeface and in the Monas hieroglyphica
7: Federico Commandino
7.1 The Life and Work of Federico Commandino and the Significance of His Commentary on the Elements
7.2 Formulation and Distribution of Genetic Definitions in Commandino´s Commentary on Euclid
7.3 The Ontological and Epistemological Status of Geometrical Motion
7.4 Commandino´s Use of Kinematic Notions in His Commentary on Book II
8: Christoph Clavius
8.1 The Life and Work of Christoph Clavius and the Significance of his Commentary on the Elements
8.2 Formulation and Distribution of Genetic Definitions in Clavius´ Commentary on Euclid
8.3 The Mechanical and Empirical Character of Genetic Definitions
8.4 The Mode of Apprehension of Geometrical Motion
8.5 Clavius´ Constructivist Interpretation of Euclidean Geometry
8.6 The Logical and Epistemological Status of Genetic Definitions
8.7 Geometrical and Mechanical Processes in the Commentary on Sacrobosco´s Sphaera and in the Commentary on Euclid
8.8 Geometrical and Mechanical Processes: The Quadratrix
8.9 Geometrical and Mechanical Processes: The Debate With Peletier on Superposition
8.10 Geometrical Motion and the Affinity of Numbers and Magnitudes in the Commentary on Book II
9: Synthesis: Continuities and Transformations in the Status of Geometrical Motion and Genetic Definitions from Fine to Clavius
9.1 Distribution and Terminology of Genetic Definitions in the Sixteenth-Century Euclidean Tradition
9.1.1 The Distribution of Genetic Definitions
9.1.2 The Designation of Motion within Genetic Definitions
9.2 The Ontological and Epistemological Status of Genetic Definitions
9.2.1 Ontology and Terminology
9.2.2 Reservations and Justifications
9.2.3 Genetic Definitions and Definitions by Property
9.3 Geometrical Motion and the Mode of Composition of Magnitudes
9.3.1 The Composition of the Continuum
9.3.2 Genetic Definitions and the Relation Between Arithmetic and Geometry
9.4 A Changing Approach to Motion and to Genetic Definitions from Fine to Clavius
9.4.1 Changes in the Epistemological Status of Genetic Definitions
9.4.2 Changes in the Ontological Status of Motion within Genetic Definitions
9.4.3 Changes in the Notion of Magnitude
10: Later Developments in the Seventeenth Century: A Cartesian Epilogue
10.1 Genetic Definitions and Instrumental Processes
10.2 Geometrical and Mechanical Processes
10.3 Geometrical Motion and the Connection Between Arithmetic and Geometry
10.4 The Constructive Approach to Geometry of Clavius and Descartes
10.5 Genetic Definitions and the Reconstruction of the World in Descartes´ Le Monde
10.6 Genetic Definitions and the Intelligibility of Geometry
Appendix
References
Index
